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A Spanning Tree Model For Khovanov Homology, Stephan Wehrli
A Spanning Tree Model For Khovanov Homology, Stephan Wehrli
Mathematics - All Scholarship
We use a spanning tree model to prove a result of E. S. Lee on the support of Khovanov homology of alternating knots.
Entire Pluricomplex Green Functions And Lelong Numbers Of Projective Currents, Dan Coman
Entire Pluricomplex Green Functions And Lelong Numbers Of Projective Currents, Dan Coman
Mathematics - All Scholarship
Let T be a positive closed current of bidimension (1,1) and unit masson the complex projective space Pn. We prove that the set Valpa(T) of points where T has Lelong number larger than alpha is contained in a complex line if alpha ≥ 2/3, and |V alpa(T ) \ L| ≤ 1 for some complex line L if 1/2 ≤ alpha < 2/3. We also prove that in dimension 2 and if 2/5 ≤ alpha < 1/2, then |V alpha (T ) \ C| ≤ 1 for some conic C.
Homology Over Local Homomorphisms, Luchezar L. Avramov, Srikanth Iyengar, Claudia Miller
Homology Over Local Homomorphisms, Luchezar L. Avramov, Srikanth Iyengar, Claudia Miller
Mathematics - All Scholarship
The notions of Betti numbers and of Bass numbers of a finite module N over a local ring R are extended to modules that are only assumed to be finite over S, for some local homomorphism phi: R -> S. Various techniques are developed to study the new invariants and to establish their basic properties. In several cases they are computed in closed form. Applications go in several directions. One is to identify new classes of finite R-modules whose classical Betti numbers or Bass numbers have extremal growth. Another is to transfer ring theoretical properties between R and S in …
The Adjoint Of An Even Size Matrix Factors, Ragnar-Olaf Buchweitz, Graham J. Leuschke
The Adjoint Of An Even Size Matrix Factors, Ragnar-Olaf Buchweitz, Graham J. Leuschke
Mathematics - All Scholarship
We show that the adjoint matrix of a generic square matrix of even size can be factored nontrivially, answering a question of G. Bergman. This note is a preliminary report on work in progress.
Explicit Lower Bounds On The Modular Degree Of An Elliptic Curve, Mark Watkins
Explicit Lower Bounds On The Modular Degree Of An Elliptic Curve, Mark Watkins
Mathematics - All Scholarship
We derive an explicit zero-free region for symmetric square L-functions of elliptic curves, and use this to derive an explicit lower bound for the modular degree of rational elliptic curves. The techniques are similar to those used in the classical derivation of zero-free regions for Dirichlet L-functions, but here, due to the work of Goldfield-Hoffstein-Lieman, we know that there are no Siegel zeros, which leads to a strengthened result.
Elliptic Curves Of Large Rank And Small Conductor, Noam D. Elkies, Mark Watkins
Elliptic Curves Of Large Rank And Small Conductor, Noam D. Elkies, Mark Watkins
Mathematics - All Scholarship
For r=6,7,...,11 we find an elliptic curve E/Q of rank at least r and the smallest conductor known, improving on the previous records by factors ranging from 1.0136 (for r=6) to over 100 (for r=10 and r=11). We describe our search methods, and tabulate, for each r=5,6,...,11, the five curves of lowest conductor, and (except for r=11) also the five of lowest absolute discriminant, that we found.
Two Theorems About Maximal Cohen--Macaulay Modules, Craig Huneke, Graham J. Leuschke
Two Theorems About Maximal Cohen--Macaulay Modules, Craig Huneke, Graham J. Leuschke
Mathematics - All Scholarship
This paper contains two theorems concerning the theory of maximal Cohen-Macaulay modules. The first theorem proves that certain Ext groups between maximal Cohen-Macaulay modules M and N must have finite length, provided only finitely many isomorphism classes of maximal Cohen-Macaulay modules exist having ranks up to the sum of the ranks of M and N. This has several corollaries. In particular it proves that a Cohen-Macaulay local ring of finite Cohen-Macaulay type has an isolated singularity. A well-known theorem of Auslander gives the same conclusion but requires that the ring be Henselian. Other corollaries of our result include statements concerning …
Transcendence Measures And Algebraic Growth Of Entire Functions, Dan Coman, Evgeny A. Poletsky
Transcendence Measures And Algebraic Growth Of Entire Functions, Dan Coman, Evgeny A. Poletsky
Mathematics - All Scholarship
In this paper we obtain estimates for certain transcendence measures of an entire function f. Using these estimates, we prove Bernstein, doubling and Markov inequalities for a polynomial P(z,w) in C2 along the graph of f. These inequalities provide, in turn, estimates for the number of zeros of the function P(z, f(z)) in the disk of radius r, in terms of the degree of P and of r. Our estimates hold for arbitrary entire functions f of finite order, and for a subsequence {nj} of degrees of polynomials. But for special classes of functions, including the Riemann zeta-function, they hold …
Quasianalyticity And Pluripolarity, Dan Coman, Norman Levenberg, Evgeny A. Poletsky
Quasianalyticity And Pluripolarity, Dan Coman, Norman Levenberg, Evgeny A. Poletsky
Mathematics - All Scholarship
We show that the graph gamma f = {(z, f(z)) in C2 : z in S} in C2 of a function f on the unit circle S which is either continuous and quasianalytic in the sense of Bernstein or C1 and quasianalytic in the sense of Denjoy is pluripolar.
Smooth Submanifolds Intersecting Any Analytic Curve In A Discrete Set, Dan Coman, Norman Levenberg, Evgeny A. Poletsky
Smooth Submanifolds Intersecting Any Analytic Curve In A Discrete Set, Dan Coman, Norman Levenberg, Evgeny A. Poletsky
Mathematics - All Scholarship
We construct examples of Cinifinity smooth submanifolds in Cn and Rn of codimension 2 and 1, which intersect every complex, respectively real, analytic curve in a discrete set. The examples are realized either as compact tori or as properly imbedded Euclidean spaces, and are the graphs of quasianalytic functions. In the complex case, these submanifolds contain real n-dimensional tori or Euclidean spaces that are not pluripolar while the intersection with any complex analytic disk is polar.
Invariant Currents And Dynamical Lelong Numbers, Dan Coman, Vincent Guedj
Invariant Currents And Dynamical Lelong Numbers, Dan Coman, Vincent Guedj
Mathematics - All Scholarship
Let f be a polynomial automorphism of Ck of degree lamda, whose rational extension to Pk maps the hyperplane at infinity to a single point. Given any positive closed current S on Pk of bidegree (1,1), we show that the sequence lamda−n(fn)*S converges in the sense of currents on Pk to a linear combination of the Green current T+ of f and the current of integration along the hyperplane at infinity. We give an interpretation of the coefficients in terms of generalized Lelong numbers with respect to an invariant dynamical current for …